Prime Ideals in Polynomial and Power Series Rings over Noetherian Domains

Ela Celikbas (), Christina Eubanks-Turner () and Sylvia Wiegand ()
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Ela Celikbas: University of Missouri, Department of Mathematics
Christina Eubanks-Turner: Loyola Marymount University, Department of Mathematics
Sylvia Wiegand: University of Nebraska–Lincoln, Department of Mathematics

A chapter in Commutative Algebra, 2014, pp 55-82 from Springer

Abstract: Abstract In this article we survey recent results concerning the set of prime ideals in two-dimensional Noetherian integral domains of polynomials and power series. We include a new result that is related to current work of the authors [Celikbas et al., Prime Ideals in Quotients of Mixed Polynomial-Power Series Rings; see http://www.math.unl.edu/~swiegand1 (preprint)]: Theorem 5.4 gives a general description of the prime spectra of the rings R [ [ x , y ] ] ∕ P , R [ [ x ] ] [ y ] ∕ Q $$R[\![x,y]\!]/P,R[\![x]\!][y]/Q$$ and R[y][​[x]​]∕Q′, where x and y are indeterminates over a one-dimensional Noetherian integral domain R and P, Q, and Q′ are height-one prime ideals of R[​[x, y]​], R[​[x]​][y], and R[y][​[x]​], respectively. We also include in this survey recent results of Eubanks-Turner, Luckas, and Saydam describing prime spectra of simple birational extensions R[​[x]​][f(x)∕g(x)] of R[​[x]​], where f(x) and g(x) are power series in R[​[x]​] such that f(x) ≠ 0 and is a prime ideal of R[​[x]​][y]—this is a special case of Theorem 5.4. We give some examples of prime spectra of homomorphic images of mixed power series rings when the coefficient ring R is the ring of integers ℤ $$\mathbb{Z}$$ or a Henselian domain.

Keywords: Commutative ring; Noetherian ring; Integral domain; Polynomial ring; Power series ring; Prime ideals; Prime spectrum; Primary 13B35; 13J10; 13A15 (search for similar items in EconPapers)
Date: 2014
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DOI: 10.1007/978-1-4939-0925-4_4

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